OK, this differential equation was technically created by me, because i need to clear my doubts.
Y'' + sqrt(X)*Y' + X^3*Y=3sin(x)
and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further.
I've never solved a second order ODE with functions as coefficients, i've always done it with constant coefficients (because those are the ones that describe oscillations), i usually solve with variation of parameters, Y=Yh+Yp where Yh=C1*Y1(x) +C2 Y2(x) calculated with the "D" operator, and then Yp=U1*Y1 + U2*Y2 and U1 and U2 are calculated integrating the division of the respective wronskians.
The only ways i have to solve these "variable coefficients" equations are via Cauchy-Euler methods (X^m) or Riccati equation knowing one solution.
I actually had to be told this is solved via variation of parameters, and i have no idea how.
Y'' +q(x)Y' + p(x) Y = g(x)
(In fact if you have any other example stored in that clarifies my doubt i'd greatly appreciate it, as long as it fits this equation above)
The Attempt at a Solution
The differential operator would be D^2 + sqrt(x)*D +X^3=0, but really i don't know how to solve for "D" (to get the general expression for Y1 and Y2) if i also have to solve for X
Thanks in advance, this might actually be a very basic doubt (it has been in my mind a long while, i know how to solve some PDEs and membrane problems with different boundary conditions but not this)